New Formulation of Statistical Mechanics using Thermal Pure Quantum States
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec December 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu NEW FORMULATION OF STATISTICAL MECHANICSUSING THERMAL PURE QUANTUM STATES
Sho Sugiura ∗ and Akira Shimizu Department of Basic Science, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo153-8902, Japan ∗ E-mail: [email protected]
We formulate statistical mechanics based on a pure quantum state, which wecall a “thermal pure quantum (TPQ) state”. A single TPQ state gives notonly equilibrium values of mechanical variables, such as magnetization andcorrelation functions, but also those of genuine thermodynamic variables andthermodynamic functions, such as entropy and free energy. Among many pos-sible TPQ states, we discuss the canonical TPQ state, the TPQ state whosetemperature is specified. In the TPQ formulation of statistical mechanics, ther-mal fluctuations are completely included in quantum-mechanical fluctuations.As a consequence, TPQ states have much larger quantum entanglement thanthe equilibrium density operators of the ensemble formulation. We also showthat the TPQ formulation is very useful in practical computations, by applyingthe formulation to a frustrated two-dimensional quantum spin system.
Keywords : statistical mechanics, pure quantum state
1. Introduction
In quantum statistical mechanics, equilibrium states are conventionally de-scribed by mixed quantum states. By contrast, recent studies have shownthe following fact.
Suppose that one prepares a pure quantum state assuperposition of the energy eigenstates whose energies lie in the energy shell[ U − ∆ U, U + ∆ U ] ( U : energy, ∆ U : energy width of o ( N )). Then, almostevery such pure state (measured by the Haar measure) gives the expec-tation values which are equal to those obtained from the microcanonicalensemble average with an exponentially small error, for any “mechanicalvariables” (See Sec. 2) such as magnetization and the correlation function.This result shows that a pure quantum state can represent a thermal equi-librium state. Motivated by this discovery, we generally call pure quantum ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu states that give the correct equilibrium value for every mechanical variablethermal pure quantum (TPQ) states. However,“genuine thermodynamic variable” such as temperature andthe thermodynamic functions cannot be calculated as the expectation val-ues of quantum-mechanical observables. In the ensemble formulation, theyare related to the number of states. Therefore, one might think it impos-sible to obtain genuine thermodynamic variables from a single
TPQ state.In this paper, however, we will show that genuine thermodynamic variablesare related to the normalization constants of appropriate TPQ states. Wepresent one example of such appropriate states, which we call the canonicalTPQ state. While the TPQ state of the previous works is specified by energy, thecanonical TPQ state is specified not by energy but by temperature. We willshow that the normalization constant of the canonical TPQ state gives thefree energy. We also present another TPQ state specified by energy, whosenormalization constant gives entropy. We call it the microcanonical TPQstate. We show that the canonical TPQ state can be constructed efficientlyfrom the microcanonical TPQ states.These results establish a new formulation of statistical mechanics, whichenables one to obtain all quantities of statistical-mechanical interest froma single realization of a TPQ state. This formulation is not only interest-ing as fundamental physics but also advantageous in practical applicationsbecause one needs only to construct a single pure state by just multiplyingthe Hamiltonian matrix to a random vector.
2. Canonical TPQ State
We consider a quantum system composed of N sites (or particles). Weassume that the dimension D of its Hilbert subspace is finite. [For par-ticle systems, D may be made finite by an appropriate truncation.] Wealso assume that for this system the ensemble formulation gives correctresults, which are consistent with thermodynamics in the thermodynamiclimit, N → ∞ . Here, we use the term “thermodynamics” in the sense ofRefs. 8,9. [This means, for example, that the entropy function is concave.]To exclude foolish operators such as N N ˆ H , we also assume that every me-chanical variable is normalized as k ˆ A k ≤ KN m where m is a constant of o ( N ) and K is a constant independent of ˆ A and N . We use quantities persite, e.g., u ≡ E/N and ˆ h ≡ ˆ H/N . The spectrum of ˆ h is assumed to bebounded, i.e., e min ≤ u ≤ e max .The canonical TPQ state | β, N i is specified by the inverse temperature ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu β and N (and possibly other variables such as magnetization, on whichwe do not explicitly write the dependence). In order to generate it, take arandom vector | ψ i ≡ X i c i | i i (1)from the whole Hilbert space. Here, {| i i} i is an arbitrary orthonormal basisset of the whole Hilbert space and { c i } i is a set of random complex numbersdrawn uniformly from the 2 D dimensional sphere, P i | c i | = D . Then, thecanonical TPQ state is given by | β, N i ≡ exp " − N β ˆ h | ψ i . (2)As we will see in the next two sections, it correctly gives both the thermo-dynamic functions and the equilibrium values of the mechanical variables.We notice that TPQ states are not the “purification” of mixed states(for details of purification, see Ref. 10), because TPQ states are pure statesin the D -dimensional Hilbert subspace, i.e., they do not require an ancilla.
3. Thermodynamic Functions and GenuineThermodynamic Variables
The free energy, which is one of thermodynamic functions, is obtained fromthe normalization constant of | β, N i as f ( β ; N ) = 1 β ln h β, N | β, N i , (3)where f ( β ; N ) ≡ − (1 /βN ) ln Z ( β, N ) is the free energy density [ Z ( β, N ) isthe partition function]. Here, we write ( β ; N ) instead of ( β, N ) in order toindicate that f ( β ; N ) converges to the N -independent one, f ( β ).Using the random matrix theory and the generalized Markov inequality,the error probability is evaluated asP (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) h β, N | β, N i exp[ − N βf (1 /β ; N )] − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ǫ (cid:19) ≤ ǫ exp[2 N β { f (1 / β ; N ) − f (1 /β ; N ) } ] , (4)where P( · ) is the probability that an event · happens. Since f (1 / β ; N ) − f (1 /β ; N ) is positive and Θ(1) from thermodynamics, the r.h.s. of in-equality (4) is Θ(1 /ǫ exp[ N ]). Therefore, a single realization of the canon-ical TPQ state almost always gives the correct thermodynamic function ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu with an exponentially small error. In another word,1 β ln h β, N | β, N i P → f ( β ) (5)where P → denotes convergence in probability.All genuine thermodynamic variables and any other thermodynamicfunctions can be obtained from f ( β ) by differentiation and the Legendretransformation.
4. Mechanical Variables
In the previous section, we have shown that the canonical TPQ state cor-rectly gives the free energy. The equilibrium values of all macroscopic quan-tities are derived from derivatives of the free energy. For mechanical vari-ables, one can also obtain their equilibrium values as the expectation valuesin the TPQ state.The expectation value of a mechanical variable ˆ A in the canonical TPQstate h ˆ A i TPQ β,N ≡ h β, N | ˆ A | β, N ih β, N | β, N i (6)gives the equilibrium value with an exponentially small error. Like the en-semble average, the expectation value is useful in many practical applica-tions.The squared average of the difference between this expectation valueand the canonical ensemble average h ˆ A i ens β,N ≡ Tr [ e − Nβ ˆ h ˆ A ] Z ( β, N ) (7)is estimated as( h ˆ A i TPQ β,N − h ˆ A i ens β,N ) ≤ h (∆ ˆ A ) i ens2 β,N + ( h A i ens2 β,N − h A i ens β,N ) exp[2 N β { f (1 / β ; N ) − f (1 /β ; N ) } ] , (8)where h (∆ ˆ A ) i ens β,N ≡h ( ˆ A − h A i ens β,N ) i ens β,N . Using the generalized Markov in-equality, we get an upper bound of the error probability asP (cid:16)(cid:12)(cid:12)(cid:12) h ˆ A i TPQ β,N − h ˆ A i ens β,N (cid:12)(cid:12)(cid:12) ≥ ǫ (cid:17) ≤ ǫ h (∆ ˆ A ) i ens2 β,N + ( h A i ens2 β,N − h A i ens β,N ) exp[2 N β { f (1 / β ; N ) − f (1 /β ; N ) } ] . (9)Since k ˆ A k < KN m (Sec. 2), the r.h.s. is Θ( N m /ǫ exp[ N ]). Therefore, asingle realization of the canonical TPQ state almost always gives the correct ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu equilibrium values of any mechanical variables with an exponentially smallerror.We have shown that the equilibrium values of both mechanical andgenuine thermodynamic variables are obtained from a single realization ofthe TPQ state. In this sense, we have established a new formulation ofstatistical mechanics based on a pure quantum state.
5. A Numerical Application
Since our formulation requires only a single pure state for each equilibriumstate, it is a powerful tool for practical applications. To illustrate this fact,we apply our formulation to a numerical computation in this section.We present the result for spin-1/2 Kagome lattice Heisenberg antifer-romagnet (KHA). This system is known to be hard to analyze because offrustration. On the ground of the numerical diagonalization of small clus-ters up to N=18, it was suggested that the specific heat of KHA would havedouble peaks at low temperature.
In Fig 1, we show our results for the specific heat. [Some detail of thecomputation will be described in Sec. 7.] The results for N = 18 a and b correspond to different shapes of the clusters. These results agree wellwith the previous results calculated by the numerical diagonalization, andshow the double peaks. However, the peak at lower temperature vanishesfor larger sizes, N = 27 and 30 [which cannot be treated by the numericaldiagonalization]. We have obtained the results for these two clusters from asingle realization of the canonical TPQ state. This suggests that the peakat lower temperature would be absent in the thermodynamic limit.Another important result is the entropy density and the free energydensity, shown in Fig. 2. We observe that there remains 45% of the totalentropy (= N ln 2) at T = 0 . J . This is a consequence of strong frustration,which makes this system hard to analyze.We emphasize again that these variables for N = 27 and 30 have beenobtained from a single realization of the canonical TPQ state. Moreover,recalling inequality (9), we can estimate the probabilistic error of the resultof the specific heat by using the result of the free energy density. The erroris estimated to be less than 1% down to T = 0 . J . Thus, our new resultsfor N = 27 and 30 are reliable enough for most purposes. ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu N = = = = Fig. 1. c vs. T of the KHA. The shapes of clusters of N = 30, 27 and 18a, 18b areshown in the right, left and in Ref. 13, respectively. s H Β ;N L f H Β ;N L - - - H Β ;N L s H Β ;N L Fig. 2. f and s vs. T for N = 30. The shape of cluster is shown in Fig. 1.
6. Microcanonical TPQ State
While we have generally defined TPQ states roughly in Sec. 1, we define itrigorously as follows. When a state | ψ i is generated from some probabilitymeasure, it is called a TPQ state if h ˆ A i ψN P → h ˆ A i ens N (10) ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu uniformly for every mechanical variable ˆ A as N → ∞ . Here, h ˆ A i ψN ≡h ψ | ˆ A | ψ i / h ψ | ψ i , h·i ens N is the ensemble average, and ‘ P → ’ denotes convergencein probability.This definition clearly shows that a single realization of the TPQ statefor sufficiently large N is enough to evaluate the equilibrium values of all mechanical variables. Among such TPQ states are the random state in theenergy shell and the canonical TPQ state. The latter has an additionalspecial property that it also gives the equilibrium values of all genuinethermodynamic variables. In this section, we present another TPQ state,called the microcanonical TPQ state, which also has this special property.Starting from the random vector | ψ i given by Eq. (1), the microcanon-ical TPQ state is defined by | k i ≡ ( l − ˆ h ) k | ψ i ( k = 0 , , , · · · ) , (11)where l is an arbitrary constant s.t. l ≥ { maximum eigenvalue of ˆ h } . Theequilibrium value of the energy density is obtained by h k | ˆ h | k ih k | k i ≡ u k . (12)More generally, the equilibrium value of a mechanical variable ˆ A is obtainedby h k | ˆ A | k ih k | k i . (13)We can show that this value, with increasing N , approaches the expectationvalue for the microcanonical ensemble of energy N u k . Thus, | k i satisfies theabove condition for a TPQ state. Furthermore, it gives the entropy density s ( u ) as 1 N ln h k | k i − kN ln( l − u k ) P → s ( u k ) . (14)Since the microcanonical TPQ state is generated by multiplying thepolynomial of ˆ h to the random vector | ψ i , it can be generated easily, e.g.,in a computer.
7. Expansion of the Canonical TPQ State
The canonical TPQ state can be decomposed as the superposition of themicrocanonical TPQ states. This decomposition enables one to performnumerical calculation efficiently. ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu We apply simple Taylor expansion to exp[
N β ( l − ˆ h ) /
2] as | β, N i = e − Nβl/ ∞ X k =0 ( N β/ k k ! | k i (15)= e − Nβl/ ∞ X k =0 R k | ψ k i . (16)where | ψ k i ≡ | k i / p h k | k i is the normalized microcanonical TPQ state, and R k ≡ p h k | k i ( N β/ k /k !.Although this Taylor expansion is the sum of infinite terms, relevant k ’sare not so many. To see the contribution of each | k i to | β, N i , we focus on R k ( > R k takes the maximum value for k such that u k isclosest to h β, N | ˆ h | β, N i . The values of R k for other k ’s decay exponentiallyfast as the corresponding u k gets further from h β, N | ˆ h | β, N i . Thus, we canefficiently generate the canonical TPQ state from a small number of themicrocanonical TPQ states, which can be numerically generated easily.The numerical results shown in Sec. 5 have been calculated using theabove relation.
8. Quantum and Thermal Fluctuations
To better understand the TPQ states, we now discuss the “quantum fluctu-ation” and “thermal fluctuation”. For concreteness, we consider the canon-ical TPQ state | β, N i and the canonical density operator ˆ ρ = e − βN ˆ h /Z .In the ensemble formulation, it is often said that a fluctuation of amechanical variable h (∆ ˆ A ) i ens ≡ h ( ˆ A − h ˆ A i ens ) i ens can be decomposedinto the quantum fluctuation h (∆ ˆ A ) i ensq and the thermal one h (∆ ˆ A ) i enst ,i.e., h (∆ ˆ A ) i ens = h (∆ ˆ A ) i ensq + h (∆ ˆ A ) i enst . (17)The thermal fluctuation, whose specific expression will be given below, isconventionally interpreted as a result of mixing many quantum states toform ˆ ρ , ˆ ρ = X n ( e − βNe n /Z ) | n ih n | , (18)where e n and | n i are eigenvalue and eigenstate, respectively, of ˆ h . Conse-quently, it is conventionally concluded that the thermal fluctuation of mostmechanical variables does not vanish at any finite temperature. ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu In the TPQ formulation, by contrast, | β, N i is a pure quantum state andtherefore does not have such “thermal fluctuation”, i.e., h (∆ ˆ A ) i TPQt = 0at all temperature. The TPQ state has only the quantum fluctuation, i.e., h (∆ ˆ A ) i TPQ = h (∆ ˆ A ) i TPQq ≡ h ( ˆ A − h ˆ A i TPQ ) i TPQ . (19)In other words, all fluctuations are included in the quantum fluctuation.We have thus found that ˆ ρ and | β, N i , which represent the same equilib-rium state, give different values of the quantum and thermal fluctuations.This does not lead to any contradiction in experimentally-observable quan-tities because h (∆ ˆ A ) i ens = h (∆ ˆ A ) i TPQ , (20)which are the only observable quantities in the above discussion. The quan-tum and thermal fluctuations, h (∆ ˆ A ) i ensq and h (∆ ˆ A ) i enst , are, separately,not observable quantities. To see this, let us write them down explicitly.We note that ρ has the following form,ˆ ρ ≡ X λ w λ | λ ih λ | , (21)where { w λ } λ is a set of positive numbers such that P λ w λ = 1, and {| λ i} λ is some set of states (which is {| n i} n in Eq. (18)). In general, ˆ A fluctuatesquantum-mechanically in each state | λ i . Hence, it may be reasonable todefine h (∆ ˆ A ) i ensq as the average of the fluctuation h λ | ( ˆ A − h λ | ˆ A | λ i ) | λ i over | λ i ’s, i.e., h (∆ ˆ A ) i ensq ≡ X λ w λ h λ | ( ˆ A − h λ | ˆ A | λ i ) | λ i . (22)This and Eq. (17) yield the thermal fluctuation as h (∆ ˆ A ) i enst = X λ w λ h λ | ˆ A | λ i − X λ w λ h λ | ˆ A | λ i ! . (23)If we take w λ = e − βNe n /Z and | λ i = | n i , we find that h (∆ ˆ A ) i enst > | λ i ’s in Eq. (21) need not be orthogonal toeach other. As a result, there are infinitely many possible choices of {| λ i} λ and { w λ } λ for the same ˆ ρ . The experimentally-observable fluctuation h (∆ ˆ A ) i ens is invariant under the change of { w λ } λ and {| λ i} λ . By contrast,both h (∆ ˆ A ) i ensq and h (∆ ˆ A ) i enst do alter under the change of { w λ } λ and {| λ i} λ . This fact clearly shows that the quantum and thermal fluctuations ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu are, separately, not experimentally-observable quantities. In other words,they are, separately, metaphysical quantities.It is instructive to consider a classical mixtureˆ ρ ′ ≡ R R X r =1 | β, N, r ih β, N, r |h β, N, r | β, N, r i (24)of many realizations | β, N, i , | β, N, i , · · · , | β, N, R i of the canonical TPQstate. Since each | β, N, r i represents the same equilibrium state, so does ˆ ρ ′ .If we define the quantum and thermal fluctuations in ˆ ρ ′ in the same wayas Eqs. (22) and (23), we find that the thermal fluctuation is exponentiallysmall for all mechanical variables. This shows that mixing many states doesnot necessarily give “thermal fluctuation”. Since the thermal fluctuation inˆ ρ ′ is negligible, we do not need to take an average over many relizations,but only need to pick up a single realization.
9. Entanglement
We have shown that the TPQ states and the density operators of thestatistical ensembles give identical results for all quantities of statistical-mechanical interest. That is, as far as one looks at macroscopic quantities,one cannot distinguish between these states. However, the TPQ states arepure quantum states while the density operators in the ensemble formu-lation (i.e., the Gibbs states) are mixed states. Therefore, the situationchanges when we look at entanglement. We discuss this point by studyingentangelement of the microcanonical TPQ state.To investigate entangelement of the TPQ state, we study its reduceddensity operator ρ q that is obtained by tracing out N − q sites. Its purity isdefined by Tr( ρ q ). Since a TPQ state is a pure quantum state, this purityis a good measure of its entanglement. The smaller the purity is, the moreentanglement the TPQ state has.In Fig. 3, we plot the minimum value of the purity (triangles N ) andthe average value of the purity of the random vector | ψ i (inverse triangles H ). It is seen that | ψ i has almost maximum (exponentially large) entan-glement. The lines are the purity of the microcanonical TPQ states withdifferent values of the energy density. It is seen that the TPQ states haveexponentailly large entanglement, and that the entanglement gets largerat higher energy, i.e., at higher temperature. This result is in marked con-trast to entanglement of the density operator of the ensemble formulation,because the latter has less entanglement at higher temperature. [For exam-ple, with increasing temperature the canonical density operator approaches ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu identity, which has no entanglement in any reasonable entanglement mea-sure.] òòòòòò ôôôôôô lowenergy high ô Average Value for Random Vector ò Minimum ValueTPQ states T r H Ρ q L Fig. 3. Purity vs. q of the 1D Heisenberg chain for N = 16. However, this is not a contradiction but a natural consequence of thenature of entanglement. The purity of ρ q is related to N -body correlationfunctions of the TPQ state. Such higher-order correlation functions repre-sent microscopic details of the TPQ state. Therefore, the great differencein entanglement between the TPQ states and the Gibbs states indicates agreat difference in microscopic details. It is not surprising that such micro-scopically completely different states give identical results for macroscopic quantities, and thus represent the same equilibrium state.
10. Conclusion
In this paper, we have established a new formulation of statistical mechan-ics based on new TPQ states. A single realization of the TPQ state givesequilibrium values of all mechanical and genuine thermodynamic variablesand thermodynamic functions, with an exponentially small error. However,the TPQ states are completely different from the Gibbs states of the ensem-ble formulation. We have illustrated this fact by showing great difference ofentanglement between them. There are many possible TPQ states, such asthe canonical TPQ state and the microcanonical one. The canonical TPQstate can be generated from the microcanonical ones, and the microcanon- ecember 19, 2013 1:22 WSPC - Proceedings Trim Size: 9in x 6in proceedings˙sugiura-shimizu ical ones can be obtained easily in a computer. This fact makes the TPQformulation advantageous in practical applications. References
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